You have no obligation to sign any non-compete agreements or the SEIYAKUSHO (which is effectively the same thing) unless you both (a) understand and (b) voluntarily agree to the terms.
For such a non-compete agreement to be valid, the eikaiwa that you are leaving you also have to offer you compensation for the hardship that non-compete agreement would cause you.
In this case, the hardships in question would relate to your right(s) to work for whomever you want to, as provided by Japan's Constitution, Civil Code, and Labor Standards Act.
This article highlights those laws:
http://generalunion.org/Joomla/index.php/laws-and-rights/1376-illegal-...***
In regards to compensation, that would be something akin to a generous severance package, or a pension fund - something of that nature.
Even if such an agreement existed and you signed it, it would only be valid for about a year, and it's unlikely that any "within X km" conditions would be upheld by a court.
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As for any non-compete clauses in the contract you held with the eikaiwa, they are pretty much always entirely null and void as they are illegal under Japanese law.
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Naturally, if you opened a new school and used resources and special techniques that only existed at the eikaiwa that you left, then the former company would have a case against you for theft of property and trade secrets.
However, it is doubtful that such techniques exist.
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If you're still worried about it, it would be best to contact the General Union about this issue and seek their advice, as they deal with this kind of issue on an almost weekly basis.
www.generalunion.org
